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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | import Mathlib
import OAI.Probability.SKBarriers.Coverage.CoverageLimit
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
open scoped Topology
theorem uniform_coverage {β : ℝ} (hβ : 1 < β) (L : ℕ → ℝ)
(hL : Tendsto L atTop atTop)
(hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) :
∃ q : ℝ, 0 < q ∧ q < 1 ∧ ∀ D : ℝ, 0 < D → ∃ D₃ : ℝ, 0 < D₃ ∧
∃ C : (n : ℕ) → Set (Disorder n), (∀ n, MeasurableSet (C n)) ∧
Tendsto (fun n => (disorderLaw n).real (C n)) atTop (𝓝 1) ∧
∀ n J, J ∈ C n → ∀ S : Finset (Config n),
finiteMass (gibbs β J) (S.filter (fun x =>
finiteMass (gibbs β J) (S.filter (fun y => q ≤ |overlap x y|)) <
Real.exp (-D₃*L n))) < Real.exp (-D*L n) := by
obtain ⟨q,hq,hq1,H⟩ := uniform_coverage_pair hβ L hL hsub
refine ⟨q,hq,hq1,?_⟩
intro D hD
obtain ⟨D₂,hD₂,hlim⟩ := H D hD
let C (n : ℕ) := (coveragePairBad n β (Real.exp (-D*L n)) (Real.exp (-D₂*L n)) q)ᶜ
have hm (n : ℕ) := measurableSet_coveragePairBad n β (Real.exp (-D*L n)) (Real.exp (-D₂*L n)) q
refine ⟨D+D₂,add_pos hD hD₂,C,(fun n => (hm n).compl),?_,?_⟩
· have Hlim := (tendsto_const_nhds (x := (1:ℝ))).sub hlim
simp only [sub_zero] at Hlim
convert Hlim using 1
funext n
dsimp only [C]
rw [measureReal_compl (hm n),probReal_univ]
· intro n J hJ S
have hpair (T : Finset (Config n)) (hT : Real.exp (-D*L n) ≤ finiteMass (gibbs β J) T) :
Real.exp (-D₂*L n) ≤ weightedOverlapMass (conditionalWeights (gibbs β J) T) q := by
have hraw : Real.exp (-D₂*L n)*(finiteMass (gibbs β J) T)^2 ≤
finiteOverlapMass (gibbs β J) T q := by
apply le_of_not_gt
intro hbad
exact hJ ⟨T,hT,hbad⟩
rw [weightedOverlapMass_conditional]
exact (le_div_iff₀ (sq_pos_of_pos ((Real.exp_pos _).trans_le hT))).mpr hraw
have Hcov := finite_coverage_of_pair_bound (gibbs β J) (fun x => (gibbs_pos β J x).le)
(Real.exp_pos (-D*L n)) (Real.exp_nonneg (-D₂*L n)) hpair S
have heq : Real.exp (-D*L n)*Real.exp (-D₂*L n) = Real.exp (-(D+D₂)*L n) := by
rw [← Real.exp_add]
congr 1
ring
rwa [heq] at Hcov
end SK.Analytic
end
end
end
end OAI
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